Lua Linear Algebra
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local linalg = {}
-- ARITHMETIC FUNCTIONS
local abs = math.abs
local floor = math.floor
local sqrt = math.sqrt
-- TRIG FUNCTIONS
local cos = math.cos
local sin = math.sin
--[[ ROW CLASS ]]--
local _rowClass = {}
local _newRow = function(values)
local instance = setmetatable({}, _rowClass)
rawset(instance, "_values", values)
return instance
end
_rowClass.__index = function(row, key)
if type(key) == "number" then
return row._values[key]
elseif key == "Length" then
return #row._values
end
error("Unrecognized key: " .. tostring(key))
end
_rowClass.__newindex = function()
error("Rows are immutable", 2)
end
_rowClass.__tostring = function(row)
local result = ""
for i = 1, row.Length do
result = result .. row[i]
if i < row.Length then
result = result .. " "
end
end
return result
end
_rowClass.__unm = function (row)
if row.Length == 1 then
return -1 * row[1]
end
local resultArray = {}
for i = 1, row.Length do
resultArray[i] = -1 * row[i]
end
return _newRow(resultArray)
end
_rowClass.__add = function (left, right)
if type(left) == "number" or type(right) == "number" then
local row = type(left) == "number" and right or left
if row.Length == 1 then
local leftScalar = type(left) == "number" and left or left[1]
local rightScalar = type(right) == "number" and right or right[1]
return leftScalar + rightScalar
end
error("Addition of a scalar to a row is undefined", 2)
end
if left.Length ~= right.Length then
error("Addition of rows of different lengths is undefined", 2)
end
if left.Length == 1 then
return left[1] + right[1]
end
local resultArray = {}
for i = 1, left.Length do
resultArray[i] = left[i] + right[i]
end
return _newRow(resultArray)
end
_rowClass.__sub = function (left, right)
if type(left) == "number" or type(right) == "number" then
local row = type(left) == "number" and right or left
if row.Length == 1 then
local leftScalar = type(left) == "number" and left or left[1]
local rightScalar = type(right) == "number" and right or right[1]
return leftScalar - rightScalar
end
error("Subtraction of a scalar to a row is undefined", 2)
end
if left.Length ~= right.Length then
error("Subtraction of rows of different lengths is undefined", 2)
end
if left.Length == 1 then
return left[1] - right[1]
end
local resultArray = {}
for i = 1, left.Length do
resultArray[i] = left[i] - right[i]
end
return _newRow(resultArray)
end
_rowClass.__mul = function (left, right)
if type(left) == "number" or type(right) == "number" then
local row = type(left) == "number" and right or left
local scalar = type(left) == "number" and left or right
if row.Length == 1 then
local leftScalar = type(left) == "number" and left or left[1]
local rightScalar = type(right) == "number" and right or right[1]
return leftScalar * rightScalar
else
local resultArray = {}
for i = 1, row.Length do
resultArray[i] = scalar * row[i]
end
return _newRow(resultArray)
end
elseif left.Length == 1 or right.Length == 1 then
if left.Length == 1 and right.Length == 1 then
return left[1] * right[1]
elseif left.Length == 1 then
return left[1] * right
else
return left * right[1]
end
end
error("Multiplication of two rows is undefined", 2)
end
_rowClass.__div = function (left, right)
if type(left) == "number" then
error("Division of a scalar by a row is undefined", 2)
elseif type(right) == "number" then
if left.Length == 1 then
return left[1] / right
else
local resultArray = {}
for i = 1, left.Length do
resultArray[i] = left[i] / right
end
return _newRow(resultArray)
end
elseif left.Length == 1 or right.Length == 1 then
if left.Length == 1 and right.Length == 1 then
return left[1] / right[1]
elseif left.Length == 1 then
error("Division of a scalar by a row is undefined", 2)
else
return left / right[1]
end
end
error("Division of two rows is undefined", 2)
end
_rowClass.__mod = function (left, right)
if type(left) == "number" then
error("Modulo of a scalar by a row is undefined", 2)
elseif type(right) == "number" then
if left.Length == 1 then
return left[1] % right
else
local resultArray = {}
for i = 1, left.Length do
resultArray[i] = left[i] % right
end
return _newRow(resultArray)
end
elseif left.Length == 1 or right.Length == 1 then
if left.Length == 1 and right.Length == 1 then
return left[1] % right[1]
elseif left.Length == 1 then
error("Modulo of a scalar by a row is undefined", 2)
else
return left % right[1]
end
end
error("Modulo of two rows is undefined", 2)
end
_rowClass.__pow = function (left, right)
if type(left) == "number" then
if right.Length == 1 then
return left ^ right[1]
else
error("Exponentiation of a scalar by a row is undefined", 2)
end
elseif type(right) == "number" then
if left.Length == 1 then
return left[1] ^ right
else
error("Exponentiation of a row by a scalar is undefined", 2)
end
elseif left.Length == 1 or right.Length == 1 then
if left.Length == 1 and right.Length == 1 then
return left[1] ^ right[1]
elseif left.Length == 1 then
error("Exponentiation of a scalar by a row is undefined", 2)
else
error("Exponentiation of a row by a scalar is undefined", 2)
end
end
error("Exponentiation of two rows is undefined", 2)
end
_rowClass.__eq = function(left, right)
if left.Length ~= right.Length then
return false
end
for i = 1, left.Length do
if left[i] ~= right[i] then
return false
end
end
return true
end
--[[ MATRIX CLASS ]]--
local _matrixClass = {}
local _newMatrix = function(rows)
local rowInstances = {}
for i = 1, #rows do
rowInstances[i] = _newRow(rows[i])
end
local instance = setmetatable({}, _matrixClass)
rawset(instance, "_rows", rowInstances)
return instance
end
_matrixClass.__index = function(mat, key)
if type(key) == "number" then
return rawget(mat, "_rows")[key]
--return mat._rows[key]
elseif key == "Shape" then
return { #rawget(mat, "_rows"), rawget(mat, "_rows")[1].Length }
elseif key == "T" then
return linalg.matrix.transpose(mat)
elseif key == "Unit" then
if mat.Shape[2] == 1 then
return linalg.vector.unit(mat)
elseif mat.Shape[1] == mat.Shape[2] then
return linalg.matrix.identity(mat.Shape[1])
else
error("Unit of a non-square matrix is undefined", 2)
end
elseif linalg.matrix[key] then
if type(linalg.matrix[key]) == "function" then
return function (...) return linalg.matrix[key](mat, ...) end
end
elseif linalg.vector[key] and mat.Shape[2] == 1 then
if type(linalg.vector[key]) == "function" then
return function (...) return linalg.vector[key](mat, ...) end
end
end
error("Unrecognized key: " .. tostring(key))
end
_matrixClass.__newindex = function()
error("Matrices are immutable", 2)
end
_matrixClass.__tostring = function(mat)
local result = ""
for i = 1, mat.Shape[1] do
result = result .. tostring(mat[i])
if i < mat.Shape[1] then
result = result .. "\n"
end
end
return result
end
_matrixClass.__unm = function(mat)
local resultRows = {}
for i = 1, mat.Shape[1] do
resultRows[i] = {}
for j = 1, mat.Shape[2] do
resultRows[i][j] = -1 * mat[i][j]
end
end
return _newMatrix(resultRows)
end
_matrixClass.__add = function(left, right)
if left.Shape[1] ~= right.Shape[1] or left.Shape[2] ~= right.Shape[2] then
error(
"Cannot add matrices of sizes (" ..
left.Shape[1] .. " x " .. left.Shape[2] .. ") and (" .. right.Shape[1] .. " x " .. right.Shape[2] .. ")",
2
)
end
local resultRows = {}
for i = 1, left.Shape[1] do
resultRows[i] = {}
for j = 1, left.Shape[2] do
resultRows[i][j] = left[i][j] + right[i][j]
end
end
return _newMatrix(resultRows)
end
_matrixClass.__sub = function(left, right)
if left.Shape[1] ~= right.Shape[1] or left.Shape[2] ~= right.Shape[2] then
error(
"Cannot subtract matrices of sizes (" ..
left.Shape[1] .. " x " .. left.Shape[2] .. ") and (" .. right.Shape[1] .. " x " .. right.Shape[2] .. ")",
2
)
end
local resultRows = {}
for i = 1, left.Shape[1] do
resultRows[i] = {}
for j = 1, left.Shape[2] do
resultRows[i][j] = left[i][j] - right[i][j]
end
end
return _newMatrix(resultRows)
end
_matrixClass.__mul = function(left, right)
local resultRows = {}
if type(left) == "number" or type(right) == "number" then
local mat = type(left) == "number" and right or left
local scalar = type(left) == "number" and left or right
for i = 1, mat.Shape[1] do
resultRows[i] = {}
for j = 1, mat.Shape[2] do
resultRows[i][j] = scalar * mat[i][j]
end
end
else
if left.Shape[2] ~= right.Shape[1] then
error(
"Cannot multiply matrices of sizes (" ..
left.Shape[1] .. " x " .. left.Shape[2] .. ") and (" .. right.Shape[1] .. " x " .. right.Shape[2] .. ")",
2
)
end
for i = 1, left.Shape[1] do
resultRows[i] = {}
for j = 1, right.Shape[2] do
resultRows[i][j] = 0
for k = 1, left.Shape[2] do
resultRows[i][j] = resultRows[i][j] + (left[i][k] * right[k][j])
end
end
end
end
return _newMatrix(resultRows)
end
_matrixClass.__div = function(mat, scalar)
if type(mat) == "number" then
error("Cannot divide a scalar by a matrix", 2)
elseif type(scalar) ~= "number" then
error("Cannot divide a matrix by a matrix", 2)
end
local resultRows = {}
for i = 1, mat.Shape[1] do
resultRows[i] = {}
for j = 1, mat.Shape[2] do
resultRows[i][j] = mat[i][j] / scalar
end
end
return _newMatrix(resultRows)
end
_matrixClass.__mod = function(mat, scalar)
if type(mat) == "number" then
error("Cannot compute modulus of a scalar by a matrix", 2)
elseif type(scalar) ~= "number" then
error("Cannot compute modulus of a matrix by a matrix", 2)
end
local resultRows = {}
for i = 1, mat.Shape[1] do
resultRows[i] = {}
for j = 1, mat.Shape[2] do
resultRows[i][j] = mat[i][j] % scalar
end
end
return _newMatrix(resultRows)
end
_matrixClass.__pow = function(base, power)
if type(base) ~= "number" and type(power) == "number" then
if base.Shape[1] ~= base.Shape[2] then
error("Cannot exponentiate a non-square matrix", 2)
end
if floor(power) ~= power or power < 1 then
error("Cannot exponentiate a matrix by a non-positive non-integer", 2)
end
-- Runs in O(n^2 log_2(n)) time
-- See https://www.hackerearth.com/practice/notes/matrix-exponentiation-1/
local resultMatrix = linalg.matrix.identity(base.Shape[1])
local curMat = base
while power > 0 do
if power % 2 == 1 then
resultMatrix = resultMatrix * curMat
end
curMat = curMat * curMat
power = floor(power / 2)
end
return resultMatrix
elseif type(base) == "number" and type(power) ~= "number" then
error("Cannot raise an arbitrary number to a matrix power", 2)
else
error("Cannot exponentiate a matrix by a matrix", 2)
--TODO (You should be able to do exactly this)
end
end
_matrixClass.__eq = function(left, right)
if left.Shape[1] ~= right.Shape[1] or left.Shape[2] ~= right.Shape[2] then
return false
end
for i = 1, left.Shape[1] do
for j = 1, left.Shape[2] do
if left[i][j] ~= right[i][j] then
return false
end
end
end
return true
end
-- MATRIX FUNCTIONS
linalg.matrix = {}
--[[**
Creates a new matrix
@param [t:array<array<number>>] rows A (m x n) array of numbers to fill the matrix with
@returns [t:(m x n) matrix] The new matrix
**--]]
linalg.matrix.new = function(rows)
return _newMatrix(rows)
end
linalg.matrix.fromColumns = function(columnVectors)
local resultRows = {}
local n = columnVectors[1]["Shape"] and columnVectors[1].Shape[1] or #columnVectors[1]
for i = 1, #columnVectors do
for j = 1, n do
if not resultRows[j] then
resultRows[j] = {}
end
resultRows[j][i] = columnVectors[i][j] * 1 --*1 to ensure scalar functionality
end
end
return _newMatrix(resultRows)
end
--[[**
Creates an identity matrix of size (n x n)
@param [t:number] n The size of the matrix
@returns [t:(n x n) matrix] The identity matrix
**--]]
linalg.matrix.identity = function(n)
local rows = {}
for i = 1, n do
rows[i] = {}
for j = 1, n do
rows[i][j] = i == j and 1 or 0
end
end
return _newMatrix(rows)
end
--[[**
Creates a matrix of all zeros of size (m x n)
@param [t:number] m
@param [t:number] n
@returns [t:(m x n) matrix] The zeros matrix
**--]]
linalg.matrix.zeros = function(m, n)
local rows = {}
for i = 1, m do
rows[i] = {}
for j = 1, n do
rows[i][j] = 0
end
end
return _newMatrix(rows)
end
--[[**
Creates a matrix of all zeros of the same size as the provided matrix
@param [t:(m x n) matrix] mat The matrix to copy the size of
@returns [t:(m x n) matrix] The zeros matrix
**--]]
linalg.matrix.zerosLike = function(mat)
return linalg.matrix.zeros(mat.Shape[1], mat.Shape[2])
end
--[[**
Creates a diagonal matrix with the values provided as the diagonal entries
@param [t:array<number>] values An n-length array whose entries will be set as the diagonal entries
@returns [t:(n x n) matrix] The resulting diagonal matrix
**--]]
linalg.matrix.diagonal = function(values)
local resultRows = {}
for i = 1, #values do
resultRows[i] = {}
for j = 1, #values do
resultRows[i][j] = i == j and values[i] or 0
end
end
return _newMatrix(resultRows)
end
--[[**
Determines whether a matrix is diagonal
Does not exclusively refer to square matrices
Refers strictly to whether all non-zero values are on the main diagonal (i.e., a_{ij} = 0 for all i, j where i ~= j)
@param [t:(m x n) matrix] mat The matrix to check
@returns [t:boolean] True if the matrix is diagonal, false otherwise
**--]]
linalg.matrix.isDiagonal = function(mat)
for i = 1, mat.Shape[1] do
for j = 1, mat.Shape[2] do
if i ~= j and mat[i][j] ~= 0 then
return false
end
end
end
return true
end
--[[**
Determines whether a matrix is upper triangular
Note that any non-square matrix will return false
@param [t:(m x n) matrix] mat The matrix to check
@returns [t:boolean] True if the matrix is upper triangular, false otherwise
**--]]
linalg.matrix.isUpperTriangular = function(mat)
if mat.Shape[1] ~= mat.Shape[2] then
return false
end
for i = 1, mat.Shape[1] do
for j = 1, mat.Shape[2] do
if i > j and mat[i][j] ~= 0 then
return false
end
end
end
return true
end
--[[**
Determines whether a matrix is lower triangular
Note that any non-square matrix will return false
@param [t:(m x n) matrix] mat The matrix to check
@returns [t:boolean] True if the matrix is lower triangular, false otherwise
**--]]
linalg.matrix.isLowerTriangular = function(mat)
if mat.Shape[1] ~= mat.Shape[2] then
return false
end
for i = 1, mat.Shape[1] do
for j = 1, mat.Shape[2] do
if i < j and mat[i][j] ~= 0 then
return false
end
end
end
return true
end
--[[**
Creates a new matrix that is the transpose of the provided matrix
@param [t: (m x n) matrix] mat The matrix to create the transpose of
@returns [t: (n x m) matrix] The transpose of mat
**--]]
linalg.matrix.transpose = function(mat)
local resultRows = {}
for i = 1, mat.Shape[1] do
for j = 1, mat.Shape[2] do
if not resultRows[j] then
resultRows[j] = {}
end
resultRows[j][i] = mat[i][j]
end
end
return _newMatrix(resultRows)
end
--[[**
Solves for e^mat
Defined as: e^A = \sum_{k=0}^{\infinity} \frac{1}{k!} A^k
Implemented in a naive way to approximate by using iterations
Runtime of O(n^3)
@param [t:(n x n) matrix] mat The matrix to use as the exponent
@param [t:number] numIterations The number of iterations to take the sum of the taylor series to
@returns The matrix exponential approximation
**--]]
linalg.matrix.expm = function(mat, numIterations)
if mat.Shape[1] ~= mat.Shape[2] then
error("Cannot find the exponential of a non-square matrix", 2)
end
if not numIterations then
numIterations = 128
end
local resultMatrix = linalg.matrix.identity(mat.Shape[1])
local curMat = resultMatrix
local factorial = 1
for i = 1, numIterations do
curMat = curMat * mat
factorial = factorial * i
resultMatrix = resultMatrix + ((1 / factorial) * curMat)
end
return resultMatrix
end
-- VECTOR FUNCTIONS
linalg.vector = {}
--[[**
Creates a new column vector
@param [t:array<number>] values The values to have for the column vector
@returns [t:(n x 1) matrix] The new column vector
**--]]
linalg.vector.new = function(values)
local rows = {}
for i = 1, #values do
rows[i] = {values[i]}
end
return _newMatrix(rows)
end
--[[**
Creates the standard basis vector i for R^n
That is, creates a vector of length n with all zeros except at index i which will have value 1
@param [t:number] i The index of e
@param [t:number] n The dimensionality of the vector
@returns [t:(n x 1) matrix] The standard basis vector e_i in R^n
**--]]
linalg.vector.e = function(i, n)
local rows = {}
for j = 1, n do
rows[j] = i == j and {1} or {0}
end
return _newMatrix(rows)
end
linalg.vector.norm = {}
--[[**
The L1 norm of a vector
sum_i{|v_i|}
@param [t:(n x 1) matrix] v The vector
@returns [t:number] The resulting value
**--]]
linalg.vector.norm.l1 = function(v)
local sum = 0
for i = 1, v.Shape[1] do
sum = sum + abs(v[i][1])
end
return sum
end
--[[**
The L2 norm of a vector
sqrt(sum_i{(v_i)^2})
@param [t:(n x 1) matrix] v The vector
@returns [t:number] The resulting value
**--]]
linalg.vector.norm.l2 = function(v)
local sum = 0
for i = 1, v.Shape[1] do
sum = sum + v[i]^2
end
return sqrt(sum)
end
--[[**
The L-infinity norm of a vector
max{v}
@param [t:(n x 1) matrix] v The vector
@returns [t:number] The resulting value
**--]]
linalg.vector.norm.linf = function(v)
local maxComponentValue = 0
for i = 1, v.Shape[1] do
local componentValue = abs(v[i][1])
if componentValue > maxComponentValue then
maxComponentValue = componentValue
end
end
return maxComponentValue
end
-- Inner product functions
linalg.vector.ip = {}
--[[**
Computes the standard dot product of two vectors
Defined as \sum_{i=0}^{n-1} v1[i] * v2[i]
@param [t:(n x 1) matrix] v1 The first vector
@param [t:(n x 1) matrix] v2 The second vector
@returns [t:number] The result
**--]]
linalg.vector.ip.dot = function(v1, v2)
if v1.Shape[1] ~= v2.Shape[1] then
error("Cannot compute the dot product for vectors of unequal dimensions", 2)
end
if v1.Shape[2] ~= 1 or v2.Shape[2] ~= 1 then
error("Cannot compute the dot product for non column vectors", 2)
end
local result = 0
for i = 1, v1.Shape[1] do
result = result + (v1[i] * v2[i])
end
return result
end
--[[**
Projects vector v onto vector space u
Defined as \sum_{i=0}^{m-1} <v, u[i]>/|u[i]|^2 * u[i]
@param [t:(n x 1) matrix] v The vector to project onto u
@param [t:array<(n x 1) matrix>] u The vector space to project v onto (can also be just one vector)
@param [t:function([(n x 1) matrix], [(n x 1) matrix])?] innerProductFunc The inner product function to use; Defaults to the dot product
@param [t:function([(n x 1) matrix])?] normFunc The norm function to use; Defaults to the L2 norm
@returns The vector projection of v onto u
**--]]
linalg.vector.project = function (v, u, innerProductFunc, normFunc)
if #u == 0 then -- u must be a single vector
u = { u }
end
if u[1].Shape[1] ~= v.Shape[1] then
error("Cannot project vectors of different dimensions", 2)
end
if u[1].Shape[2] ~= 1 or v.Shape[2] ~= 1 then
error("Expected two vectors, not matrices", 2)
end
if not innerProductFunc then
innerProductFunc = linalg.vector.ip.dot
end
if not normFunc then
normFunc = linalg.vector.norm.l2
end
local result = linalg.matrix.zerosLike(v)
for i = 1, #u do
result = result + (innerProductFunc(v, u[i]) / normFunc(u[i])^2) * u[i]
end
return result
end
--[[**
Gets the unit vector with the same direction as the provided vectr
@param [t:(n x 1) matrix] v The vector with the appropriate direction
@param [t:function?] normFunc The function to use as the norm; Defaults to the L2 norm
@returns [t:(n x 1) matrix] The unit vector
**--]]
linalg.vector.unit = function(v, normFunc)
if not normFunc then normFunc = linalg.vector.norm.l2 end
return v / normFunc(v)
end
--[[**
Creates a matrix that rotates a vector about an arbitrary vector
Only works for 3 dimensions
@param [t:(n x 1) matrix] v The vector to rotate about (should be a unit vector)
@param [t:number] theta The angle to rotate by (in radians)
@returns [t:nxn matrix] The resulting linear operator
**--]]
linalg.vector.createArbitraryAxisRotationMatrix = function(v, theta)
if v.Shape[1] ~= 3 then
error("Cannot create rotation matrix about arbitrary unit vector other than in R^3", 2)
end
local costheta = cos(theta)
local sintheta = sin(theta)
local versine = 1 - costheta
local u = v.Unit
return _newMatrix({
{versine * u[1] ^ 2 + costheta, (versine * u[1] * u[2]) - (sintheta * u[3]), (versine * u[1] * u[3]) + (sintheta * u[2])},
{(versine * u[1] * u[2]) + (sintheta * u[3]), versine * u[2] ^ 2 + costheta, (versine * u[2] * u[3]) - (sintheta * u[1])},
{(versine * u[1] * u[3]) - (sintheta * u[2]), (versine * u[2] * u[3]) + (sintheta * u[1]), versine * u[3] ^ 2 + costheta}
})
end
--[[ GENERAL FUNCTIONS ]]--
--[[**
Creates an orthonormal basis for a dim-dimensional inner product space
@param [t:array<(n x 1) matrix>] u The list of matrices to add to the basis (will be converted to unit vectors) (can be a single vector instead of an array)
@param [t:number?] epsilon The minimum norm value for a vector to count to be added to the basis; Defaults to 0.01
@param [t:function([(n x 1) matrix], [(n x 1) matrix])?] innerProductFunc The inner product function to use; Defaults to the dot product
@param [t:function([(n x 1) matrix])?] normFunc The norm function to use; Defaults to the L2 norm
@returns [t:array<(n x 1) matrix>] An orthonormal basis that includes the unit vectors of the original u
**--]]
linalg.gramSchmidt = function (u, epsilon, dim, innerProductFunc, normFunc)
if #u == 0 then -- u must be just a single vector
u = { u }
end
if not epsilon then
epsilon = 0.01
end
if not dim then
dim = u[1].Shape[1]
end
if not innerProductFunc then
innerProductFunc = linalg.vector.ip.dot
end
if not normFunc then
normFunc = linalg.vector.norm.l2
end
-- Normalize all elements in u
local newU = {}
for i = 1, #u do
newU[i] = u[i].Unit
end
u = newU
for i = 1, dim do
local eiRows = {}
for j = 1, dim do
eiRows[j] = { i == j and 1 or 0 }
end
table.insert(u, _newMatrix(eiRows))
end
local basisVectors = { u[1].Unit }
for i = 1, #u do
local potentialBasisVector = u[i] - linalg.vector.project(u[i], basisVectors, innerProductFunc, normFunc)
if normFunc(potentialBasisVector) >= epsilon then
table.insert(basisVectors, potentialBasisVector.Unit)
end
if #basisVectors == dim then
break
end
end
return basisVectors
end
return linalg
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